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This is a request for a list of examples of problems (or other mathematical situations) that are not initially of algebro-geometric nature, but can be solved or understood by using algebraic geometry.

Here are some applications that are not of the kind sought:

  1. Diophantine equations or other problems whose basic data are specified in algebraic terms, or have an immediate translation into such terms.

  2. GAGA or reduction to finite characteristic arguments, but applied to problems that are clearly already within (or very near) the algebro-geometric sphere, involving varieties or moduli spaces, or cohomology of such spaces.

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Any question that asks for a "big list" should also explain carefully how it's going to help you in your research. Are you trying to learn algebraic geometry and looking for motivation? Are you going to teach algebraic geometry and looking for examples? Why are you asking this question? – Loop Space Jun 23 '10 at 20:54
Not sure if this counts: if $\Gamma$ is any group and $(V_i, \rho_i)$ are two finite-dimensional semisimple linear representations of $\Gamma$ in characteristic 0, then $\rho_1 \otimes \rho_2$ is semisimple. The only known proof for general $\Gamma$ (according to Serre) is by introducing algebraic geometry via Zariski closures of each $\rho_i(\Gamma)$ in ${\rm{GL}}(V_i)$: prove they're reductive (perhaps disconnected!), and apply facts about linear algebraic groups to reduce the problem to its easy analogues for semisimple Lie algebras and finite groups. Serre is very fond of this argument. – BCnrd Jun 23 '10 at 21:00
@BC: thanks. @Andrew: the FAQ, HowToAsk and material they link to don't mention any extra need for a "big-list" question to explain the asker's individual interest in the subject. Indeed, doing so would (or could) be extramathematical or needlessly personal, though it can add value sometimes. IMO it is enough that the math in a question, considered on its own, seems likely to interest people enough to gain answers or upvotes. The latter have occurred 4 and 7 times respectively within nine hours, so apparently the question was interesting enough without personal details on research etc. – T.. Jun 24 '10 at 5:43
@Andrew: your objection seems much more relevant to the question about "Math for dinner". – Boyarsky Jun 24 '10 at 11:18
I didn't realize upon reading the FAQ that everyone was under an obligation to convince Andrew that her/his question is vital for her/his research! Mercifully, this fearsome rule is rarely enforced in practice. – Victor Protsak Jun 28 '10 at 13:39

11 Answers 11

Explicit theta function representations of soliton solutions to the completely integrable models (the work pioneered by Dubrovin, Matveev, Novikov, Krichever, McKean...).

In the simplest case of the KdV equation, solitons are obtained from a singularization of the corresponding hyperelliptic curve. The kdV equation was originally derived to describe waves in shallow water and presumably had nothing to do with hyperelliptic curves.

Edit. And eventually the connection worked in the opposite direction as well: a solution of Shottky's problem which exploited the integrability of the Kadomtsev–Petviashvili equation.

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Great answer. Is the Sato infinite-dimensional Grassmannian (on which KdV becomes a linear flow) also understood algebro-geometrically these days, or is it mostly the construction of soliton solutions using algebraic curves and moduli spaces (Krichever et al) that provides the connection? – CFZ Jun 23 '10 at 20:33
@ CFZ: Thank you for the comment. Unfortunately, I'm not a specialist in soliton equations (let alone the algebro-geometric aspects of the theory) and I only learned a bit about the connection when I was an undergraduate. So I'd better let someone with more knowledge comment on this. – Andrey Rekalo Jun 23 '10 at 20:50
In papers on geometric Langlands one sometimes sees discussion ind-schemes (inductive limits) especially including infinite Grassmannians. What I was wondering is whether the KdV-related Grassmannian is understood using algebraic geometry constructs per se (sheaves, cohomology, etc) in addition to having a nice flow on a homogeneous space. Since Langlands stuff seems to be half the discussion on this site I was hoping someone will know! – CFZ Jun 23 '10 at 21:23
One could peruse Dubrovin lectures on the subject: – mathphysicist Jun 23 '10 at 22:32
@ mathphysicist: Many thanks for the reference! – Andrey Rekalo Jun 23 '10 at 22:35

The construction of certain Steiner systems is a good example.

A $(p, q, r)$ Steiner system is a collection of $q$-element subsets $A$ (called blocks) of an $r$-element set $S$ such that every $p$-element subset of $S$ is contained in a unique element of $A$. Good examples come from considering as blocks the set of hyperplanes in $\mathbb{A}^n$ or $\mathbb{P}^n$ over a finite field. For example, $\mathbb{A}^2$ over $\mathbb{F}_3$ gives a $(2, 3, 9)$ Steiner system: it contains $9$ ($\mathbb{F}_3$-rational) points, and let the blocks be the lines, each of which consists of $3$ points. Then any $2$ points are contained in a unique line. This is the unique $(2, 3, 9)$ Steiner system.

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Error-correcting codes are a similar example, of constructing combinatorial objects from varieties over finite fields. – CFZ Jun 23 '10 at 19:56
(I mean Goppa codes and the later refinements, nor error correcting codes in general; most codes don't come from algebraic geometry.) – CFZ Jun 23 '10 at 19:58
I hadn't seen Goppa codes (before wikipedia'ing them just now)---this is quite nice. – Daniel Litt Jun 23 '10 at 20:07
Generally, for any interesting combinatorial object there is a desire to construct it (or $q$-deform / quantize it, find coverings and automorphisms, etc) using geometry over finite fields. For example, the outer automorphism of $S_6$ can be understood using geometry mod 5, and other special finite groups can be realized as (linear or projective) algebraic groups over finite fields. – CFZ Jun 23 '10 at 20:24
Interesting--I have heard rumors of a story connecting the outer automorphism of $S_6$ to the Mathieu group $M_{12}$, which is the automorphism group of the unique Steiner system $(5, 6, 12)$. A possible connection? – Daniel Litt Jun 23 '10 at 20:33

Given a convex polytope whose facets are simplices, define the f-vector by f_i = the number of i-dim faces. Which vectors of integers are f-vectors? A list of conditions was conjectured, proven sufficients by direct construction of enough polytopes, and proven necessary by applying hard Lefschetz to the (rationally smooth) toric variety associated to the dual polytope. (A combinatorial proof came later.) See Fulton's book on toric varieties.

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Wow! This is amazing. – Daniel Litt Jun 23 '10 at 22:19
I agree. This should also be a nice motivation for starting to learn algebraic geometry (or at least toric geometry). – Martin Brandenburg Jun 24 '10 at 9:04
Could you give a reference for the combinatorial proof? (I didn't know about that.) – Torsten Ekedahl Jun 25 '10 at 18:11
To be completely honest, the toric proof only works for $\textit{rational}$ convex polytopes (which seems like totally insignificant technicality until you learn that there are non-rational polytopes not combinatorially equivalent to any rational ones), whereas combinatorial intersection cohomology, although certainly inspired by it, works in complete generality. – Victor Protsak Jun 28 '10 at 6:45
I thought that the proof of the strong Lefschetz for combinatorially defined intersection cohomology was proven by a reduction to the simplicial case which in turn reduces to the rational case. Hence the proof is not independent of algebraic geometry. – Torsten Ekedahl Jun 28 '10 at 7:50

There exists a unique map from permutations of 1,2,3... that move finitely many numbers, to their "Schubert polynomials" in ${\mathbb Z}[x_1,x_2,...]$, satisfying the following recursion: $S_{id} = 1$, and if $w(i) > w(i+1)$, then $S_{w r_i} = (S_w - r_i \cdot S_w) / (x_i - x_{i+1})$. (Here $r_i$ switches $i$ and $i+1$, or $x_i$ and $x_{i+1}$.)

It's not too hard to prove that these are polynomials, form a basis of the polynomial ring, have positive coefficients, and much else. The nonobvious theorem is that the structure constants (expanding a product of two basis elements in the basis) are positive. The only proofs known of this are geometric.

(This is perhaps a lame example, in that the motivation for Schubert polynomials was geometric -- they represent the classes of Schubert varieties in the cohomology rings of flag manifolds.)

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Possibly less lame positivity example(s): Kazhdan-Lusztig polynomials are positive, and the product of Kazhdan-Lusztig basis elements (in the Hecke algebra) expands positively as a sum of Kazhdan-Lusztig basis elements. – Alexander Woo Jun 24 '10 at 0:06
Haiman's proof of the n! conjecture using the Hilbert scheme is another application to combinatorics. For the K-L coefficients, the first proofs were by intersection cohomology (Beilinson-Bernstein in the early 1980's), but haven't combinatorial proofs been given since that time? – CFZ Jun 24 '10 at 1:46
My understanding is that combinatorial proofs are only known in some special cases, e.g. in type A where one of the permutations avoids some pattern. I'm not sure if positivity is even known in general in the non-crystallographic case. – Qiaochu Yuan Jun 24 '10 at 14:06

Of course, the Cayley Hamilton Theorem is not really hard, and there are many many proofs of it. But you have to admit that, at least when you first step into linear algebra, it's rather surprising that it's enough to proof the theorem for diagonal matrices (which is a very short calculation). Because then you can derive it for diagonalizable matrices, which are dense with respect to the Zariski Topology (assuming w.l.o.g. that the ground field is algebraically closed). The latter is because every non-empty open subset is dense, a rather strange but here very useful property.

The same procedure applies to other polynomial identites in linear algebra, for example that the characteristic polynomials of $AB$ and $BA$ coincide.

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Victor, it is not backwards: semisimplicity is implied by the separability of the characteristic polynomial, which is to say that its geometric zero set is disjoint from that of its derivative, or equivalently the resultant of the pair is non-vanishing. The latter is a Zariski-open locus (even hypersurface complement) within the irreducible space of all $n \times n$ matrices (over a field) and so is Zariski-dense if it has a geometric point. A suitable diagonal matrix over an infinite extension provides such a point. So semisimplicity holds at the generic point, which is what we need. QED – Boyarsky Jun 25 '10 at 10:20
Victor, your suggested proof of Cayley-Hamilton looks like the standard incorrect proof: how can you specialize $\lambda$ to be a matrix (such as $A$) when the computation rests on it being an element of the commutative coefficient ring over which the matrices live? Please clarify what I am misunderstanding. – Boyarsky Jun 25 '10 at 16:09
1 If you want to say that CH holds at the generic point then I agree, and generic semisimplicity is then a corollary which isn't needed in the proof of CH itself. I don't see a direct proof of Zariski density of diag matrices w/distinct eigenvalues (geom pts), because some form of CH is needed to relate them to the condition that the char polynomial has distinct roots. 2 Let $S$ be the commutant of $A$ in $M_n$, then $S[\lambda]\subset M_n[\lambda]$ contains $X=A-\lambda I_n$ and $adj X$ and spec'n is a unique ring hom $\phi:S[\lambda]\to M_n$ that is identity on $S$ and $\phi(\lambda)=A.$ – Victor Protsak Jun 25 '10 at 22:00
Victor, how about the following: if the char. poly. splits with distinct roots, then we get an eigenline for each eigenvalue and hence have diagonalized the matrix with distinct eigenvalues. The converse is clear, and we haven't used Cayley-Hamilton. I think that addresses your #1. For #2, very nice! (Alas, it seems too slick to be used when teaching Cayley-Hamilton to inexperienced undergraduates.) – Boyarsky Jun 25 '10 at 22:05
I don't understand your comments about generic points. Nothing like that is needed in the proof. @Victor: I get the message, I also don't think that the proof in my answer is the most natural one. But it's just awesome ;-). – Martin Brandenburg Jun 28 '10 at 10:28

I think this is an example: In Arnold's intro PDE book, he discusses Maxwell's theorem about spherical functions being expressible in terms of derivatives of 1/r. The appendix gives topological and algebraic geometry interpretations. Unfortunately I don't know enough about it to give a better description.

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The polynomial method is a powerful, albeit somewhat mysterious and fragile, tool in extremal combinatorics, being used for instance in Dvir's proof of the Kakeya conjecture over finite fields:

This is currently the only known proof of the full conjecture. Previously to Dvir's work, algebraic geometry methods did not feature prominently in the prior partial results.

A related method is Stepanov's method to count points in algebraic varieties over finite fields, though this is clearly a question which was already well within the purview of algebraic geometry to begin with.

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A really nice example is the (unpublished) work of Larsen and Pink on the "rough" classification of subgroups of $\mbox{GL}_n(k)$. Here's a link:

In one sentence, the idea is to study these subgroups by looking at their "effective Zariski closures", whereupon techniques of algebraic geometry may be brought to bear on the problem.

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Why was this 1998 paper never published? Has anyone apart from the authors read it in detail to vouch for its correctness? (I know, this criterion isn't satisfied by quite a few published papers...) – Boyarsky Jun 26 '10 at 2:08
I think other people know more about this than me. I don't think there's any serious doubt as to its correctness (I've read parts of it in detail myself). I heard that one of the referees objected to a use of the word "generic", which wasn't the standard one in algebraic geometry, and so things dragged a bit and then somehow it never got published. The paper is very well known and admired in the group theory community I believe. – Ben Green Jun 26 '10 at 12:48

A classic result in 3-manifold topology is Neuwirth's conjecture, which states that the fundamental group of a knot complement is a free product of two proper subgroups amalgamated along a free group. This was proven by Culler and Shalen using the algebraic geometry of representation varieties of 3-manifold groups into $SL_2 C$. Since this is an affine variety, one may associate at least two ideal points (the non-triviality of the representation variety follows from Thurston's geometrization theorem). Associated to these ideal points is an action on a Bass-Serre tree, and then a technique of Stallings associates to this a separating (for at least one ideal point) surface with boundary, and the desired amalgamated product.

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Define the Ramanujan $\tau$-function from $\mathbb N \rightarrow \mathbb Z$ as the Fourier coefficients of the $\Delta$ function; i.e.,

$$ \sum_{n\geq 1}\tau(n)q^n=q\prod_{n\geq 1}(1-q^n)^{24} .$$

This is a pure number-theoretic function. Now the Ramanujan conjecture says that

$$|\tau(p)| \leq 2p^{11/2} $$

for every prime $p$, which is also a purely number theoretic statement.

Pierre Deligne proved it as a consequence of the Weil conjectures.

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There are several papers by Michael Atiyah where he studies partial differential equations and distributions by methods of algebraic geometry - especially the Hironaka resolution of singularities. See eg Resolution of Singularities and Division of Distributions or the two articles with Bott and Garding about Lacunas for Hyperbolic Differential Operators with Constant Coefficients.

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